5.6.8 let f be a uniformly continuous function on a set e. show that if {xn} is a cauchy sequence in e then {f(xn)} is a cauchy sequence in f(e). show that this need not be true if f is continuous but not uniformly continuous.
Proof. (1): To prove {f(xn)} is a Cauchy sequence just need to prove ∀ > 0, ∃N, s.t.,∀n, m > N,
have |f(xn) − f(xm)| < . Since f is uniformly continuous on set E, thus ∀ > 0, ∃δ > 0, s.t.,
∀x, y ∈ E, if |x − y| < δ,then |f(x) − f(y)| < .as {xn} is a Cauchy sequence, then ∃N, s.t.,
∀n, m > N, |xn − xm| < δ, thus |f(xn) − f(xm)| < which proves {f(xn)} is a Cauchy sequence.
(2): for example f(x) = 1
x
, x ∈ (0, 2) which is continuous but not uniformly continuous. {
1
n
} is
a Cauchy sequence, however, {f(xn)} does not converge which proves that it is not a Cauchy
sequence.